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Limits without log

  1. Mar 19, 2010 #1
    1. The problem statement, all variables and given/known data
    Find the limit as n--> infinity of (1-2/n)^n


    2. Relevant equations

    We know (1+1/x)^x --> e as n--> infinity

    3. The attempt at a solution

    I worked it out as e^(-2) using log but I can't get it out using the fundamental limit above. I know it's the square of (1-1/x)^x (where we let x=n/2), just I don't know how to show that (1-1/x)^x --> 1/e. If you could let x |--> -x somehow I'd get the desired result using the limit laws but I'm not sure that's allowed.
     
  2. jcsd
  3. Mar 19, 2010 #2

    Mark44

    Staff: Mentor

    Let n = -2x. This makes your limit
    [tex]\lim_{-2x \to \infty} (1 + \frac{1}{x})^{-2x}[/tex]

    With a bit of adjustment you can use the limit you know.
     
  4. Mar 19, 2010 #3
    but won't the parameter go to -infinity so we can't equate (1+1/x)^x to e?
     
  5. Mar 19, 2010 #4

    Mark44

    Staff: Mentor

    As it turns out,
    [tex]\lim_{x \to -\infty} (1 + \frac{1}{x})^x~=~e[/tex]

    Can you use this fact?
     
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